Bounded Inflation and Heavy Evolution Introduction

Let us still consider the problem of regulating a dynamical economy ( i ) for almost all t

1

0, z'(t) = c(z(t),p(t)) where p(t) E P ( z ( t ) ) where P : K

-

Z associates with each commodity z the set P ( z ) of feasible prices (in general commodity-dependent) and c : Graph(P) ^I+ X is the change function.

For simplicity, we take for allocation set the domain K := Dom(P) of p31.

We have seen in the preceding section that viable price functions (which provide viable solutions z(t) E K := Dom(P)) are the ones obeying the regulation law

- - -

310r we replace P by its restriction to K.

where

But, as we have seen in the simple example of the Introduction, heavy evolutions (evolutions with constant price) have to be switched instantate- neously when the boundary of the set of allocations is reached.

In order to avoid the use of such impulses (which are however observed in real economic systems in times of crisis), we may impose a bound'on the velocity of the prices, i.e., a constraint of the form

and start anew the study of the evolution of the dynamical economy under this new constraint.

Therefore, in this section, we are looking for systems of differential equa- tions or of differential inclusions governing the evolution of both viable com- modities and prices, so that we can look for

- heavy solutions, which are evolutions where the prices evolve with minimal velocity

- punctuated equilibria, i.e., evolutions in which the price jj remains constant whereas the commodity may evolve in the associated viability cell, which is the viability domain of z ^H c(z, p),

The idea which allows us to achieve these aims is quite simple: we dif- ferentiate the regulation law.

This is possible whenever we know how to differentiate set-valued maps.

Hence the first section is devoted to the definition and the elementary prop- erties of the contingent derivative32 DF(z, y) of a set-valued map F : X

-

^Y

a t a point (2, y) of its graph: By definition, its graph is the contingent cone to the graph of F at (z, y). We refer to Chapter 5 of SET-VALUED ANALYSIS for further information on the differential calculus of set-valued maps.

In the second section, we differentiate the regulation law and deduce that (ii) for almost all t

>

^{0, p'(t) E DUK} (z(t), p(t))(c(z(t),p(t))) whenever the viable price p(-) is absolutely continuous,

This is the second half of the system of differential inclusions we are looking for.

32We set D f ( z ) := D f(z, f (2)) whenever c i s single-valued. When f is FrCchet differ- entiable at z, then D f(z)(v) = fl(z)v is reduced t o the usual directional derivative.

Observe that this new differential inclusion has a meaning whenever the commodity-price pair (z(-),p(-)) remains viable in the graph of

nK.

Fortunately, by the very definition of the contingent derivative, the graph of IIK is a viability domain of the new system (i), (ii).

Unfortunately, as soon as viability constraints involve inequalities, there is no hope for the graph of the contingent cone, and thus, for the graph of the regulation map, to be closed, so that, the Viability Theorem cannot apply.

A strategy inspired from economic motivations to overcome the above dif- ficulty is to bound the inflation, for instance

(iii) for almost all t

2

0, Ilp'(t)ll

5

(p(z(t),p(t))

In this case, we shall look for graphs of closed set-valued maps

II

contained in Graph(P) which are viable under the system of differential inclusions. We already illustrated that in the simple economic example of Introduction.

Such set-valued maps II are solutions to the system of first-order partial differential inclusions

satisfying the constraint

Since we shall show that such closed set-valued maps II are all contained in the regulation map IIK, we call them subregulation maps associated with the economy with bounded inflation i ) , iii). In particular, there exists a largest subregulation map denoted II9.

In particular, any single-valued w : K ^o

Z

with closed graph which is a solution to the partial differential inclusion

satisfying the constraint

provides "feedback prices" regulating smooth allocations of the dynamical econ- omy.

Set-valued and single-valued solutions t o these partial differential in-

We call it the metaregulation law associated with the subregulation map

n.

This is how we can obtain smooth viable commodity-price solutions t o our pricing problem by solving the system of differential inclusions (i), (iv).

Section 3 is devoted t o selection procedures of dynamical feedbacks, which are selections g(., .) of the metaregulation map

They can be obtained through selection procedures introduced in the preced- ing section.

Naturally, under adequate assumptions, we shall check in Section 4 that Michael's Theorem implies the existence of a continuous dynamical feedback.

But under the same assumptions, we can take as dynamical feedback the minimal selection go(.,') defined by l l ! ? O ( z , ~ ) ( ( = minv~~II(x,p)(c(x,p))

IIvII,

which, in general, is not continuous.

However, we shall prove that this minimal dynamical feedback still yields smooth viable price-commodity solutions to the system of differential equa- tions

z'(t) = c(z(t),p(t)) & ~ ' ( t ) = gO(z(t),p(t))

called heavy viable solutions, (heavy in the sense of heavy trends.) They are the ones for which the price evolves with minimal velocity.

Heavy viable solutions obey the inertia principle: "keep the prices constant as long as they provide viable allocations".

Indeed, if zero belongs t o DIT(z(tl), p(tl))(c(z(tl), p(tl ))), then the price will remain equal t o p(tl) as long a s for t 2 tl, a solution z(-) t o the differen-

tial equation z'(t) = c(z(t), p(tl)) satisfies the condition 0 E DII(z(tl ), p(tl))(c(x(tl), p ( t l ) ) ) . If a t some time t i , p(tf) is a "punctuated equilibrium", then the solution

enters the viability cell associated t o this price and may remain in this

viability cell forever33 and the price will remain equal to this punctuated equilibrium.

3.1 Contingent Derivatives

By coming back to the original point of view proposed by Fermat, we are able to geometrically define the derivatives of set-valued maps from the choice of tangent cones to the graphs, even though they yield very strange limits of differential quotients.

Definition 3.1 Let F : X ^.u Y be a set-valued map from a normed space

Im Dokument Static and Dynamic Issues in Economic Theory. III. Dynamical Economies (Seite 75-79)